Hyperbolic Volterra equations of convolution type in Sobolev spaces

TL;DR

This paper investigates the solvability of hyperbolic Volterra integro-differential equations in Sobolev spaces, extending the theory to include equations relevant in viscoelasticity within a Hilbert space framework.

Contribution

It generalizes the solvability results for hyperbolic Volterra equations in Sobolev spaces, applicable to viscoelasticity models in Hilbert spaces.

Findings

Established correct solvability in weighted Sobolev spaces

Extended integro-differential equations theory in Hilbert spaces

Applicable to viscoelasticity models

Abstract

We study the correct solvability of an abstract integro-differential equations in Hilbert space generalizing integro-differential equations arising in the theory of viscoelastisity. The equations under considerations are the abstract hyperbolic equations perturbed by the terms containing Volterra integral operators. We establish the correct solvability in the weighted Sobolev spaces of vector-valued functions on the positive semiaxis.